Critical Inequalities and Their Role in Forming the Set Ψ1 in L-Function Theory

Table of Links

1. Abstract & Introduction
2. Notation and outline of the proof
3. The set Ψ1
4. Zeros of L(s, ψ)L(s, χψ) in Ω
5. Some analytic lemmas
6. Approximate formula for L(s, ψ)
7. Mean value formula I
8. Evaluation of Ξ11
9. Evaluation of Ξ12
10. Proof of Proposition 2.4
11. Proof of Proposition 2.6
12. Evaluation of Ξ15
13. Approximation to Ξ14
14. Mean value formula II
15. Evaluation of Φ1
16. Evaluation of Φ2
17. Evaluation of Φ3
18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums

References

3. The set Ψ1

Let ν(n) and υ(n) be given by

respectively. It is easy to see that

Lemma 3.1. Assume (A) holds. Then

Proof. Let

which has the Euler product representation

For σ ≥ σ0 > 0, by checking the cases χ(p) = ±1 and χ(p) = 0 respectively, it can be seen that

and

the implied constant depending on σ0. Thus φ(s) is analytic for σ > 1/2 and it satisfies

for σ ≥ σ1 > 1/2, the implied constant depending on σ1. The left side of (3.2) is

Lemma 3.2. Assume (A) holds. Then we have

Proof. As the situation is analogous to Lemma 3.1 we give a sketch only. It can be verified that the function

is analytic for σ > 1/2 and it satisfies

for σ ≥ σ1 > 1/2, the implies constant depending on σ1. Also, one can verify that

This completes the proof.

Lemma 3.3. For any s and any complex numbers c(n) we have

and

Proof. The first assertion follows by the orthogonality relation; the second assertion follows by the large sieve inequality.

Let

By (3.1) we may write

By Cauchy’s inequality and the first assertion of Lemma 3.3 we obtain

Thus we conclude

Lemma 3.4. The inequality

Write

Assume that (A) holds. By Cauchy’s inequality, the second assertion of Lemma 3.2 and Lemma 3.1,

Thus we conclude

Lemma 3.5 Assume that (A) holds. The inequality

Let

Assume that (A) holds. By Cauchy’s inequality, the first assertion of Lemma 3.2 and Lemma 3.1,

Thus we conclude

Lemma 3.6. Assume that (A) holds. The inequality

We are now in a position to give the definition of Ψ1: Let Ψ1 be the subset of Ψ such that ψ ∈ Ψ1 if and only if the inequalities (3.4), (3.5) and (3.6) simultaneously hold.

Proposition 2.1 follows from Lemma 3.4, 3.5 and 3.6 immediately.